How does the \(\operatorname{BA}\)  hierarchy fit within this framework? Given two linear minions \({{M}}\)  and \({{N}}\)  such that \({{M}}\)  is conic, we define their semi-direct product \({{M}}\ltimes {{N}}\)  as a new linear minion that is essentially designed in a way to make the corresponding relaxation stronger than the relaxations associated with \({{M}}\)  and \({{N}}\) . In particular, one easily checks that \({{M}}\ltimes {{N}}\)  is in fact a conic minion. Hence, this operation can be viewed as a standard way of making a given linear minion \({{N}}\)  conic; or, in other words, a way to make a given relaxation locally consistent.
The minion introduced in [1]} corresponding to the first level of \(\operatorname{BA}\)  is the semi-direct product of the conic minion \({{Q}_{\operatorname{conv}}}\)  and the linear minion \({{Z}_{\operatorname{aff}}}\) . We can then capture algebraically the \(\operatorname{BA}\)  hierarchy as follows.
